1-s2.0-S1001605815604577-main

68

2015,27(1):68-75 DOI: 10.1016/S1001-6058(15)60457-7

Numerical simulation of rotating arm test for prediction of submarine rotary derivatives* PAN Yu-cun (潘雨村)1,2, ZHOU Qi-dou (周其斗)2, ZHANG Huai-xin (张怀新)1 1. State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 2. Department of Naval Architecture, Naval University of Engineering, Wuhan 430033, China, E-mail: [email protected] (Received June 18, 2013, Revised November 6, 2014) Abstract: The numerical method is used for predicting the rotary-based hydrodynamic coefficients of a submarine. Unsteady RANS simulations are carried out to numerically simulate the rotating arm test performed on the SUBOFF submarine model. The dynamic mesh method is adopted to simulate the rotary motions. From the hydrodynamic forces and moments acting on the submarine at different angular velocities, the rotary derivatives of the submarine can be derived. The computational results agree well with the experimental data. The interaction between the sail tip vortex and the cross flow in the hull boundary layer is discussed, and it is shown that the interaction leads to the “out-of-plane” loads acting on the submarine. Key words: submarine maneuverability, hydrodynamic coefficients, rotating arm test, dynamic mesh

Introduction The rotary derivative of a submarine is one of the

most important hydrodynamic parameters, and it sig- nificantly affects the maneuverability and the dynamic stability of the vehicle. The rotating arm experiment is an effective method to determine the rotating-related hydrodynamic coefficients. The submarine model is fixed to a rotating arm, while the radius of rotation and the angular velocity can be adjusted systematica- lly. The transverse force and the yawing moment ac- ting on the hull at different angular velocities are mea- sured. Consequently, by analyzing these results, the rotary derivatives of the model can be derived. More- over, it is believed that the rotary derivatives derived from the rotating arm test are generally more accurate than those from the planar motion mechanism (PMM) experiment.

* Project supported by the National Natural Science Foun- dation of China (Grant No. 11272213). Biography: PAN Yu-cun (1980- ), Male, Ph. D. Candidate, Lecturer Corresponding ahthor: ZHANG Huai-xin, E-mail: [email protected]

With the advances of the computational fluid dy- namics (CFD), much effort is put in the numerical si- mulations for manoeuvering purposes[1-6]. The CFD method can be considered as a “numerical rotating arm basin”, which can be used to predict the forces and moments directly from the flow field around a submarine model. Gregory[7] put a deformed body in a rectilinear flow to investigate the flow separation over a body of revolution in a steady turning state. The total force on the curved body can account for the case of rotation within 5% of deviation. The results for the moments see a difference of 20% for / = 5R L and a difference of 100% for / = 3R L . Sung et al.[8] simula- ted the flow around a turning submarine named ONR Body 1. The deviations between the computed forces and moments and those of experiments were within 20%. Zhang et al.[9] performed a computational study of the Series 58, SUBOFF and DRDC STR bare hulls undergoing steady turning maneuvering. They found that the rotation increases the lateral force and reduces the yawing moment relative to a hull in a pure tran- slation at equivalent drift angles. Hu and Lin[10] com- puted the hydrodynamic coefficients of an autono- mous underwater vehicle SMAL01 based on an added momentum source method.

In some investigations[8-10], the submarine rota-

69

tion was considered in two ways: first, at the inlet of the computational domain, the incident flow varies li- nearly with the constant turning rate from the center of rotation, second, the Navier-Stokes (NS) equations were adjusted to a body-fixed frame of reference. In doing so, the unsteady flow can be treated as a steady problem. The vehicle could remain stationary in the control volume. It is convenient for meshing and nu- merical computations. But there are some drawbacks: both the NS equations and the turbulence models should be modified in a rotating coordinate system. Further, in the presence of the background rotation, the solid wall and far field boundary conditions should be treated carefully, which can significantly affect both accuracy and convergence. The gradients of the Cartesian velocity are set to zero, and the pressure is obtained by a non-reflecting condition as suggested by Sung[8]. Since in the physical experiment, the hull ro- tates through undisturbed fluid, it is intuitive to adopt the inertial frame of reference in solving the NS equa- tions for the submarine motion. The artificial pressure gradient[11] is avoided in simulating the flow field in such a way, and results might be closer to the physical reality. The difficulty of this method lies in the mo- ving boundary. To overcome this difficulty, the dyna- mic mesh method or the overset mesh method was purposed. Carrica et al.[12] simulated the flow field of a surface combatant in the steady turning state and the PMM test using an overset grid method. Pan et al.[13] applied the dynamic mesh method to simulate the PMM experiment performed on the SUBOFF subma- rine model.

The aim of the present study is to explore the po- ssibility of developing a numerical method to predict the rotary derivatives of a submarine. The virtual rota- ting arm basin experiments are conducted using the unsteady RANS solver in an inertial reference system, and the moving boundary of the vehicle is taken into account by the dynamic mesh method. The resultant forces and moments are then post–processed to com- pute the rotating-related coefficients. The flow field around the rotating submarine is also discussed. 1. Numerical model 1.1 Description of the model test

The target studied in this paper is the SUBOFF model. The entity model is a body of revolution, with- out bow planes, with a sail, two horizontal planes and two vertical rudders, and a ring wing supported by four struts in an “X” configuration. The overall length of the SUBOFF model is 4.356 m, while the length between the perpendicular edges is 4.261 m, and the maximum diameter is 0.508 m.

Generally, the 6 DOF motion of the submarine is described using two coordinate systems. The first is a

right-handed, body-fixed coordinate system, with its origin O at a point 2.013 m aft of the forward per- pendicular edge on the hull centerline, which is pre- scribed as the center of the buoyancy (CB). The -x axis points upstream, the -y axis points starboard and

the -z axis points downward.

Fig.1 Principal earth-fixed and body-fixed coordinate systems

The second coordinate system, an inertial refere- nce frame, is used to define the motions of the first coordinate system, as shown in Fig.1. In this earth- fixed coordinate system, the position of the vehicle’s CB is then expressed in , , coordinates. The

orientation of the body-fixed coordinate system is de- scribed by Euler angles (yaw) ), (pitch) , (roll) .

The origin E of the earth-fixed coordinate sys- tem is located at the center of the virtual basin, and the model is mounted to a virtual rotating arm which re- volves about the axis E . In Fig.1, viewing from

above, the model rotates clockwise with a steady an- gular velocity, r , that implies that the model turns to the starboard. The modelʼs -y axis coincides with the

arm, while -x axis and -z axis are kept normal to the arm. Thus, the transverse velocity component of the model’s CB is always zero and the longitudinal velo- city component is equal to its linear speed. 1.2 Governing equations

Numerical simulations are performed with the CFD software Ansys Fluent. The flow around the ve- hicle is modeled using the incompressible, Reynolds averaged Navier-Stokes (RANS) equations:

( )= 0i

ixu

(1)

( )+ = +i i i

j i i jj i j j

u uPu F u ut x x x xu

(2)

where t is the time, iu are the time averaged velo-

70

city components in Cartesian coordinates ( = 1,2,ix i

3) , is the fluid density, iF are the body forces,

P is the time averaged pressure, is the viscous

coefficient, iu are the fluctuating velocity compone-

nts. The finite volume method is employed to discre-

tize the governing equations with the second-order up- wind scheme. The semi-implicit method for the pre- ssure-linked equations (SIMPLE) is used for the pre- ssure-velocity coupling. In order to allow the closure of the time averaged Navier-Stokes equations, various turbulence models were introduced to provide an esti-

mation of the Reynolds stress tensor i ju u .

In Sungʼs work[8], the predictions about a subma- rine model ONR Body-1 in a steady turning state with different turbulence models were compared. Moreover, the simulations of the SUBOFF model at the incide- nce was also studied. In most cases, the realizable

-k model provides better agreements with the ex- perimental results. Therefore, this model is also cho- sen in the present paper. 2. Computational method 2.1 Computational domain and boundary conditions

The computational domain is a cylindrical volu- me without an inlet or outlet. The outer boundaries are two body lengths, 2L , away from the hull. These boundaries are either of constant radius or of constant depth so that the flow is nominally tangential to the flow direction. A static pressure opening condition is applied on these boundaries, which allow for the en- trainment without a predetermined flow direction, so that the flow can either enter or leave the boundary as dictated by the solution[9]. The no-slip boundary con- dition is applied to the hull surfaces.

Fig.2 Surface grid of the model 2.2 Mesh definition

The fluid domain is divided into three regions: an inner region, an outer region, and an intermediate layer between them. In the inner region, the multi- block hexahedral grids are used and refined in critical regions such as the boundary layer, the trailing edge

of the sail and the control surface, as shown in Fig.2. The coarse hexahedral elements are used in the outer region, which are adequate for the far field. The inter- mediate layer is covered by unstructured tetrahedral grids, which can be conveniently re-meshed in the case of the boundary deformation, as shown in Fig.3. Such a hybrid mesh strategy allows a fine grid resolu- tion to resolve the viscous flow features near the hull, while saves the computation cost in the outer region where the flow has small variations. The geometry modeling and the grid generation are done by using the Gambit software.

Fig.3(a) Meshes in three sub-regions

Fig.3(b) The near-body meshes in three sub-regions

To numerically simulate the moving boundary of the vehicle, the dynamic mesh method is used. In the transient simulation, the outer domain remains statio- nary in space, while the inner region containing the SUBOFF model rotates around the -z axis. It should be noted that the mesh in the inner region remains lo- cked in a position relative to the motion of the vessel. Hence, the mesh of the intermediate layer is deformed to accommodate the motion of the inner region. The new node locations are updated at each time step ac- cording to the calculation of the user defined fun- ction (UDF), while the overall mesh topology is main- tained. 2.3 Numerical experimental parameters

In the rotating arm experiment, the model rotates

71

at a constant linear speed, while the radius of rotation and the angular velocity are adjusted accordingly. The non-dimensional angular velocity, r , is related to the length of the model, L , and the linear velocity U as follows:

=rLrU

(3)

the linear velocity is, in turn, related to the angular ve- locity, r , and the towing radius, R , by

=U rR (4) therefore, the non-dimensional angular velocity is ty- pically expressed as

=LrR

(5)

As is known, one of the difficulties in the rota-

ting arm tests is the determination of the forces at = 0r . Since the linear velocity is constant, if the an-

gular velocity approaches zero, the radius would be infinity. Due to the restriction on the diameter of a physical basin, the non-dimensional angular velocity r generally ranges from 0.2 to 0.5.

Fig.4(a) Lateral forces coefficient Y at different angular ve-

locities

Fig.4(b) Yawing moment coefficient N at different angular

velocities

As for the numerical rotating arm basin, the re- striction on the arm length is not a primary obstacle. In this paper, the linear velocity of the model keeps as 4 m/s (with the Reynolds number of 1.693×107 based on the vehicle length), while the radius of rotation va- ries from 60 m to 20 m, the corresponding non-dimen- sional angular velocity is about 0.07-0.2.

The hydrodynamic force and moment acting on the turning model are non-dimensionalized as follows:

2 2

=12

YYU L

, 2 3

=12

NNU L

(6)

Figure 4 shows the lateral force and the yawing

moment varying with the angular velocity. The values of the rotary derivatives are determined from the slo- pes of the curves about the force and moment coeffi- cients versus the angular velocity at = 0r , as follo- ws:

=0

=rr

YYr

, =0

=rr

NNr

(7)

3. Results and discussions 3.1 Verification and validation

In the present study, the verification and valida- tion (V&V) method proposed by Stern et al.[14,15] is performed for the rotary derivatives rY and rN . The

benchmark data are the results of Roddy’s PMM ex- periment[16].

Table 1 Results for rY and rN with different grids

Mesh rY

(CFD) rY

(Exp.) rN

(CFD) rN

(Exp.)

Fine

1( )S 0.005269

0.005215

0.004948

0.004444Medium

2( )S 0.005208 0.004923

Coarse

3( )S 0.005387 0.004891

3.1.1 Grid convergence

To inspect the grid convergence, a series of sys- tematically refined grids are generated. In view of the limited computational resource, different refinement ratios are adopted in different regions. Since the near- wall grid has a dominant influence on the precision of the CFD simulation, in the inner region, the grid refi-

nement is conducted with a ratio of 2 in three dire-

72

ctions, while in the intermediate layer and the outer

region, the refinement ratio is less than 2 . The fine, medium and coarse grids have 1.341107, 7.15106

and 4.31106 grid cells, respectively. The +y value,

i.e., the non-dimensional normal distance to the wall from the first grid, is approximately 30 for the fine grid, 43 for the medium grid and 60 for the coarse grid.

Table 1 summarizes the rotary derivatives rY and rN computed with different grids. The solutions

corresponding to the three grids are presented as 1S ,

2S and 3S . The differences of the solutions for

medium-fine and coarse-medium solutions are defined as

21 2 1= S S , 32 3 2= S S

The convergence ratio GR is defined as

21

32

=GR

Table 2 shows the grid convergence ratio GR ,

the order of accuracy GP , the correction factor GC ,

and the grid uncertainty GU .

Table 2 Grid convergence verification of rY and rN

GR GP GC 1(% )GU S

rY 0.342 – – 1.70

rN 0.781 0.712 0.280 4.40

Since 0 1GR , the monotonic convergence is

obtained for rN . Thus, the generalized Richardson

extrapolation[14] can be used to estimate the grid un-

certainty GU . The estimates for the error G and the

order of accuracy GP are defined as

21= =1G GG RE p

Gr

,

32

21

ln

=ln( )G

G

Pr

The correction factor is defined as

1=

1

G

Gest

pG

G pG

rCr

And, gestp is an estimate for the limiting order of ac-

curacy as the spacing size tends to zero. In the present paper, the expected order of the flow solver is the se- cond. Here, = 0.28GC indicates that the leading-order

term over-predicts the error. The grid uncertainty GU

can be estimated as

= + (1 )G G RE G REG GU C C

For rY , the convergence ratio GR is negative,

which implies that rY is oscillatorily converged.

Under this condition, the grid uncertainty is estimated simply by bounding the error based on the oscillation maximums US and minimums LS , i.e.,

1

= ( )2G U LU S S

The values of the grid uncertainty GU (1.7% 1S ,

4.4% 1S ) for rY and rN are reasonable in view of the

overall number of grid points used. 3.1.2 Time step convergence

For the time step convergence study, three diffe- rent time steps are used to obtain solutions with =tD

0.003318, 0.004694, and 0.006637 (non-dimensionali- zed with /L V ).

Table 3 Results for rY and rN with different time steps

rY

(CFD) rY

(Exp.) rN

(CFD) rN

(Exp.)

Fine

1( )S 0.005311

0.005215

0.004935

0.004444Medium

2( )S 0.005269 0.004948

Coarse

3( )S 0.005172 0.004965

Table 4 Time step convergence verification of rY and rN

TR TP TC 1(% )TU S

rY 0.444 2.342 1.252 0.97

rN 0.765 0.774 0.308 2.04

Tables 3 shows the solutions of the three time

steps, Table 4 shows the verification parameters for the time step study. The definitions of the time step convergence parameters are similar to the grid conver- gence parameters. The subscript “G” indicates “Grid”,

73

and the subscript “T” indicates “Time step”. For both rY and rN , the TR values satisfy

0 1TR . That is, the solution converges monotoni-

cally with the time step refining. Hence, the numerical verification is demonstrated for this case. Reasonably small levels for the time step uncertainty TU (0.97%,

2.04%) are predicted for the coefficients rY and rN , respectively. The uncertainty levels are smaller than those in the grid study, which suggests that the system is less sensitive to the time step size.

Table 5 Validation of rY and rN

(% )E D (% )DU D (% )SNU D (% )VU D

rY 1.840 10 1.99 10.20

rN 11.05 10 5.46 11.39

3.1.3 Validation

The validation parameters are summarized in Table 5. =E D S is the comparison error, i.e., the difference between the experimental data D and the simulated value S . DU is the uncertainty of the ex-

perimental data. In Roddyʼs experiment[16], an uncer- tainty value of 10% is suggested for the experimental data of rotary derivatives, i.e., = 10%DU D . The si-

mulation numerical uncertainty SNU is defined as

22 2= + +SN I G TU U U U . Here, the iterative uncertai-

nty IU is at least two orders of magnitude less than

the grid uncertainties, and is considered to be negligi-

ble, therefore, 22= +SN G TU U U .

The validation uncertainty VU is defined as

2 2= +V SN DU U U . Since the errors E are smaller

than the uncertainties VU , the rotary derivatives rY and rN are validated at the uncertainty levels of

10.20% and 11.39%, respectively. From the above results, the rotary derivatives

derived from the CFD agree well with those from the experiments[16]. It can be preliminarily concluded that the CFD calculation is reliable for estimating the rota- ry derivatives of a fully appended submarine by simu- lating the rotating arm test. 3.2 Flow field

The longitudinal distribution of the lateral force Y on the model in a typical turning state ( = 0.15)r

is shown in Fig.5. It is necessary to point out that the lateral velocity varies along the hull. In the present study, the angle of drift at the CB is set to zero. How- ever, the local drift angle ( ) varies with its longi-

tudinal location over the entire length of the submari- ne, as shown in Fig.6. Here, is the distance between the local point and the forward perpendicular edge. At the forward perpendicular edge, where = 0 , the local drift angle approaches its negative peak, and this angle increases as its location moves toward the stern. Thus, the CB is the critical location where the drift angle changes from negative to positive.

Fig.5 Longitudinal distribution of lateral force Y on the

SUBOFF model

Fig.6 Drift angle variation in longitudinal locations

As is known, the sail is located between / =L 0.2 and 0.3. In Fig.5, for the region in front of the sail, that is, the location moves ahead from / = 0.2L to

/ = 0L , the absolute value of the cross flow velocity increases, thus the absolute value of the lateral force increases accordingly. However, when / 0.03L , the windward projected area sharply decreases, thus the absolute value of Y drops.

At the point of / = 0.2L , the sail creates a high lift, which brings the lateral force Y to its mini- mum value. For the points about / = 0.89L and

/ = 0.95L , they correspond to the control surfaces and the duct. These appendages have relatively large windward projected areas, and they have relatively larger lateral velocities. Thus, it is not surprising that the maximum lateral force Y is created here.

To help understand the mechanism that generates the forces and moments on the body, the flow field

74

around the submarine in = 0.15r is studied.

Fig.7 The flow field around the SUBOFF submarine

Figure 7 shows the velocity contours around the submarine as it travels in the rotating arm test simula- tion. The Q criterion, i.e., based on the positive se-

cond invariant of the velocity gradient tensor, is used to define the type and the location of the vortex core[17]. Here, the vortical structures are shown as iso- surfaces of = 50Q .

During the submarine’s maneuvering, the sail en- counters a lateral flow and works as a finite span wing. The lift develops on the sail, as a consequence, a st- rong vortex is shed from the sail tip, and it is conve- cted downstream along the hull. The sail tip vortex rolls the low velocity wake flow from the sail, and in- teracts with the boundary layer in the cross flow arou- nd the hull.

Fig.8 Flow field at / = 0.23L for SUBOFF model at =r

0.15

Another typical feature of the flow field is the horseshoe vortex, which is generated around the sail- hull junction, and it is rolled and elongated as it trave- ls along the side face of the sail.

Figure 8 shows an example of the velocity and pressure fields in a cross plane, looking toward the model’s bow. The plane of / = 0.232L is just be- hind the leading edge of the sail. Here, a pair of hor- seshoe vortices are shown at both sides of the sail-hull junction. The local drift angle is negative, that is, the starboard of the body is expected to be on the wind- ward side. Hence, the lateral flow creates a pressure difference between the starboard and the port, what is more, the presence of the sail mightily amplifies this difference. Therefore, the negative lateral force ( )Y

is produced. However, for the aft body, the direction of the

cross flow changes its direction. The plane of / =L 0.58 is a cross section behind the CB. The portside of the body becomes the windward side, and the lateral force induced by the pressure difference points to the starboard. (i.e., Y ). The horseshoe vortices are also affected by the cross flow velocity and move towards the starboard.

Fig.9 Flow field at / = 0.58L for SUBOFF model at =r

0.15

Another point of interest is that the hull in the ho- rizontal motion is acted by a vertical force, called the “out-of-plane” loads. In Fig.9(a), the direction of the

75

vortical flow shedding from the sail tip is consistent with that of the cross flow in the hull boundary layer. And the merging of the two flows results in a higher velocity on the upper hull in comparison to that on the lower hull. According to the Bernoulliʼs theorem, the relatively lower pressure on the upper hull surface is produced, as shown in Fig.9(b), which consequently results in an upward normal force ( )Z and a bow

pitch down moment. 4. Conclusions

A numerical procedure for the prediction of the rotary derivative of a submarine is proposed, based on solving the unsteady incompressible Reynolds-avera- ged Navier-Stokes equations in an inertial frame of re- ference system. The force and the moment on the SUBOFF model during the rotating arm test are su- ccessfully calculated. The prediction of the rotary coe- fficients of the submarine model enjoys an acceptable level of accuracy.

The longitudinal distribution of the lateral force on the model in a turning state is investigated. The local drift angle varies with its longitudinal location, which significantly affects the lateral force. From the CFD solution, the plots of the velocity and pressure distributions at several longitudinal locations are crea- ted and analyzed.

Another noticeable phenomenon is the intera- ction between the vortex shed from the sail and the hull boundary layer. For a submarine in the horizontal rotating arm test, this interaction results in a upward force on the aft hull and consequently a bow pitch down moment. This flow physics is illustrated and ex- plained with the help of the computed flow field. References [1] PAN Zi-ying, WU Bao-shan and SHEN Hong-cui. Re-

search of CFD application in engineering estimation of submarine maneuverability hydrodynamic forces[J]. Journal of Ship Mechanics, 2004, 8(5): 42-51(in Chinese).

[2] SIMONSEN C. D., STERN F. RANS maneuvering si- mulation of Esso Osaka with rudder and a body-force propeller[J]. Journal of Ship Research, 2005, 49(2): 98-120.

[3] LI Ying-hua, WU Bao-shan and ZHANG Hua. Resea- rch on application of prediction for unsteady maneuve- ring motion of underwater vehicle by dynamic mesh te- chnique in CFD[J]. Journal of Ship Mechanics, 2010, 14(10): 1100-1108(in Chinese).

[4] PANG Yong-jie, YANG Lu-cun and LI Hong-wei et al. Approaches for predicting hydrodynamic characteristics of submarine objects[J]. Journal of Harbin Enginee- ring University, 2009, 30(8): 903-908(in Chinese).

[5] PHILLIPS A. B., TURNOCK S. R. and FURLONG M. Evaluation of manoeuvring coefficients of a self-prope- lled ship using a blade element momentum propeller model coupled to a Reynolds averaged Navier Stokes flow solver[J]. Ocean Engineering, 2009, 36: 1217- 1225.

[6] RACINE B. J., PATERSON E. G. CFD-based method for simulation of marine-vehicle maneuvering[C]. 35th AIAA Fluid Dynamics Conference and Exhibit. Toronto, Ontario, Canada, 2005.

[7] GREGORY P. Flow over a body of revolution in a steady turn[D]. Doctoral Thesis, Melbourne, Australian: University of Melbourne, 2006.

[8] SUNG C. H., JIANG M. Y. and RHEE B. et al. Valida- tion of the flow around a turning submarine[C]. The 24th Symposium on Naval Hydrodynamics. Fukuoka, Japan, July, 2002.

[9] ZHANG J. T., MAXWELL J. A. and GERBER A. G. et al. Simulation of the flow over axisymmetric submarine hulls in steady turning[J], Ocean Engineering, 2013, 57: 180-196.

[10] HU Z. Q., LIN Y. Computing the hydrodynamic coeffi- cients of underwater vehicles based on added mome- ntum sources[C]. The 18th International Offshore and Polar Engineering Conference. Vancouver, Canada, 2008, 451-456.

[11] RAY A., SINGH S. and SESHADRI V. CFD as a con- cept design tool for manoeuvring hydrodynamics of Un- derwater vehicles[C]. Warship 2010: Advanced Tech- nologies in Naval Design and Construction. London, UK, 2010.

[12] CARRICA P. M., WILSON R. V. and NOACK R. W. et al. A dynamic overset, single-phase level set approa- ch for viscous ship flows and large amplitude motions and maneuvering[C]. The 26th Symposium on Naval Hydrodynamics. Rome, Italy, 2006.

[13] PAN Yu-cun, ZHANG Huai-xin and ZHOU Qi-dou. Numerical prediction of submarine hydrodynamic coe- fficients using CFD simulation[J]. Journal of Hydro- dynamics, 2013, 25(1): 840-847.

[14] STERN F., WILSON R. V. and COLEMAN H. W. et al. Comprehensive approach to verification and validation of CFD simulationsPart 1: Methodology and procedu- res[J]. Journal of Fluids Engineering, 2001, 123(4): 793-802.

[15] WILSON R. V., STERN F. and COLEMAN H. W. et al. Comprehensive approach to verification and validation of CFD simulationsPart 2: Application for RANS si- mulation of a cargo/container ship[J]. Journal of Flui- ds Engineering, 2001, 123(4): 803-810.

[16] RODDY R. F. Investigation of the stability and control characteristics of several configurations of the DARPA SUBOFF model (DTRC model 5470) from captive- model experiments[R]. DTRC/SHD 1298-08, 1990, 1- 108.

[17] LIU Zhi-hua., XIONG Ying and WANG Zhan-zhi et al. Numerical simulation and experimental study of the new method of horseshoe vortex control[J]. Journal of Hydrodynamics, 2010, 22(4): 572-581.